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Stochastic Differential Geometry, a Coupling Property, and Harmonic Maps
Author(s) -
Kendall Wilfrid S.
Publication year - 1986
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-33.3.554
Subject(s) - mathematics , brownian motion , harmonic map , bounded function , manifold (fluid mechanics) , mathematical analysis , differential geometry , euclidean space , property (philosophy) , riemannian manifold , stochastic differential equation , coupling (piping) , curvature , pure mathematics , differential (mechanical device) , harmonic , euclidean geometry , geometry , physics , mechanical engineering , philosophy , statistics , epistemology , quantum mechanics , engineering , thermodynamics
A Riemannian manifold has the Brownian coupling property if two Brownian motions can be constructed on it, with arbitrary initial points, and such that they are sure to meet at some time. While Euclidean space has this property, simply‐connected manifolds with negative curvature bounded away from zero do not. Such a coupling is then not possible even if Γ‐martingales of bounded dilatation are used rather than Brownian motions. A generalised little Picard theorem for harmonic maps is proved using these results.